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The amount of blows needed to be heard:
The Biblical requirement:
According to Biblical law, one is only is required to hear nine blows, which is made up of three sets of a Tekiah-Teruah-Tekiah [תר”ת תר”ת תר”ת]. With the passage of time, the question arose regarding the identity of the sound of the Teruah with there being three sound possibilities. Due to this, the Sages instituted that one is to blow three sets of all the possible sounds of Teruah in order to avoid any question as to whether one has fulfilled his obligation. Hence one is to blow three sets of Tekiah-Shevarim Teruah-Tekiah [תשר”ת], and afterwards three sets of Tekiah-Shevarim-Tekiah [תש”ת], and afterwards three sets of Tekiah-Teruah-Tekiah [תר”ת]. Thus, the total sum of sounds needed to heard is 30 sounds.
The Rabbinical requirement and additional customs:
*See below, towards end of article for an elaboration of these two subjects!
The blows of Musaf: The Sages instituted that a Minyan is to blow the Shofar within the Musaf prayer, as will be explained in Chapter 4 Halacha 1. Various customs exist regarding the amount of sounds one is to blow during the Musaf prayer and regarding whether they are to be blown in the silent prayer of Musaf or only in the Chazan’s repetition. The Chabad custom is to blow thirty sounds in both the silent and repetition prayer of Musaf. See Chapter 4 Halacha 5 for the full details of this subject!
The blows following Musaf: It is customary for a Minyan to blow the Shofar also after the conclusion of the prayer of Musaf. Various reasons are given for this custom and hence consequently various customs exist regarding the amount of sounds one is to blow. The Chabad custom is to blow ten blows after Musaf, during Kaddish, and another thirty at the conclusion of Davening, after Tehillim. See Chapter 4 Halacha 15 for the full details of this subject!
When are the Biblical blows sounded?
[The original custom was to blow the Shofar before Shacharis until] the Sages instituted to blow the Shofar within Musaf. The custom of all Jewry however eventually became [if there is a Minyan] to blow the Biblical set of blows before Musaf and then again in Musaf. The blows before Musaf are sounded after the Torah reading.
Must the 30 blows be sounded in any specific order?
As long as there was a total of 30 sounds blown, three times the set of Tashrat, three times the set of Tashat and three times the set of Tarat one fulfills his obligation. See Halacha 9 for the full details on this subject!
From the letter of the law one is Biblically required to blow a total of thirty blows which include three sets each of Tashrat, Tashat, Tarat. The Sages instituted to blow a certain amount of blows also in the Musaf prayer. The Chabad custom is to blow a total of 130 blows throughout the Davening. The following is the list of amounts and their areas of sounding:
Thirty after Kerias Hatorah [to fulfill the Biblical command]
Thirty during the silent Shemoneh Esrei of Musaf [to fulfill the Rabbinical command].
Thirty by Chazan’s repetition of Musaf [to fulfill the Rabbinical command].
Ten in the middle of the Kaddish recited directly after Musaf, prior to Tiskabel [to complete the 100 blows].
Thirty after Davening [to be Yotzei anyone who did not yet hear the sounds, and to confuse the Satan].
If one is Davening without a Minyan, when is he to blow the Shofar, before or after Musaf?
He is to blow the Shofar before Musaf. See chapter 4 Halacha 6!
If one is Davening without a Minyan how many sounds is he to blow?
The custom is to blow a total of 100 sounds. The additional sounds [beyond the thirty sounds blown before Musaf] are to be sounded after Musaf. See chapter 4 Halacha 6!
How many sounds must one blow when blowing for others, such as on Mivtzaim?
Initially one is to blow 30 sounds for each and every individual group, consisting of Tashrat 3x, Tarat 3x and Tashat 3x. If one is unable to do so then he may blow 10 blows consisting of one set each of Tashrat-Tashat-Tarat [without a blessing]. Some opinions however rule that even initially one may blow only 10 blows in order to be able to merit as many Jews as possible in the Mitzvah. Others however rule that one does not fulfill the Mitzvah at all unless he blows a minimum of 30 blows each time.
Blowing three times Tashrat with a Tnaiy/stipulation: Some Rabbanim of Anash rule it is permitted for one to blow three sets of only Tashrat for the sake of meriting other Jews, so long as he makes the following stipulation prior to the blowing: The blower is to have in mind prior to commencing the blows that the validity of the Shevarim-Teruah is dependent on the identity of the correct blow, as is revealed before G-d, and whatever sound is not the correct blow is considered like the voice of an animal, and is not being done for the sake of a Mitzvah. In this way it is considered as if he heard three sets of each Tashrat, Tashat and Tarat.
Sparks of Kabala:
Each set of blows has a Segula to subjugate another aspect of our evil inclination. The blowing of Shofar of Meyushav [after Kerias Hatorah] nullifies the evil inclination for idol worship. The blowing of Shofar of Chazaras Hashtaz of Musaf nullifies the evil inclination for murder. The blowing of Shofar of the silent Musaf prayer nullifies the evil inclination for promiscuity and Giluiy Arayos. The ten sounds blown during Kaddish of after Musaf nullify the evil inclination for Lashon Hara.
How many sounds is one to blow during Musaf:
Various customs exist regarding the amount and the type of sounds blown during Musaf. The [widespread Ashkenazi] custom is to blow a single set of תשר”ת [4 blows] at the end of each blessing. Others are accustomed to blow a single different set at the end each blessing. They blow one set of תשר”ת for the Kingship blessing, one set of תש”ת for the Remembrance blessing, and one set of תר”ת for the Shofros blessing. Others are accustomed to blow three sets of one of the three groupings by each blessing. Thus they blow three times תשר”ת for the Kingship blessing, three times תש”ת for the Remembrance blessing, and three times תר”ת for the Shofros blessing. Others are accustomed to blow one set of each of the three groupings [for a total of three sets] by each blessing. Thus they blow תשר”ת/תש”ת/תר”ת by each one of the three blessings. Practically the best of the above customs is like the last custom mentioned. [In the Siddur Admur rules like this latter opinion and so is the Chabad custom.]
Various customs exist in this regard. The best custom to follow is to blow תשר”ת/תש”ת/תר”ת by each one of the three blessings and so is the Chabad custom.
Blowing after Musaf:
[It is customary to blow the Shofar after the conclusion of the prayer of Musaf. Various customs exist regarding how many sounds one is to blow.] Some communities have the custom after the prayer to blow a long Teruah [“Teruah Gedola”] without a Tekiah. Other communities have the custom to repeat thirty blows [after Musaf]. Other communities have the custom to blow [a total of] 100 blows [throughout the day]. They hence blow after Davening whatever amount of blows needed [to reach the total of 100] after calculating the amount of blows that were sounded in Meyushav [after Kerias Hatorah] and Meumad [Musaf]. [Practically the widespread custom is like this latter opinion, to blow the remaining sounds for a total of 100 blows and by the last blow they sound a Tekiah Gedola [and not a Teruah Gedola]. The Chabad custom is to follow all three customs mentioned above. We blow ten blows [Tashrat Tashat Tarat] after Musaf, during Kaddish, [to complete the total of 100 blows, as rules the 3rd custom]. We do not sound a Tekiah Gedola by the last Tekiah of the ten sounds. We then blow another thirty at the conclusion of Davening, after Tehillim [as rules the second custom]. The last Tekiah sounded at the end of these thirty blows is a Tekiah Gedola [as rules the first custom regarding a Teruah Gedola].]
The Chabad custom is to blow ten blows after Musaf and another thirty at the conclusion of Davening, after Tehillim. The last Tekiah of the thirty is sounded as a Tekiah Gedola.
Must one repeat the sound if he made a mistake in the blows of after Musaf?
No, unless one knows that there is someone present who did not yet fulfill his obligation
 Admur 590:1-2
 Background of the Biblical obligation to blow 9 blows: Biblically one is only required to hear nine blows on Rosh Hashanah [תר”ת תר”ת תר”ת]. This is learned from the fact the Torah states the word Teruah three times, twice regarding Rosh Hashanah and one time regarding Yom Kippur of the year of Yovel. The oral tradition received that these three areas are juxtaposed and hence it is considered as if the word Teruah was said three times by both Rosh Hashanah and Yom Kippur of Yovel. Now every Teruah must have a simple sound precede it and come after it. This sound is called a Tekiah. This obligation to blow a Tekiah before the Teruah is learned from the verse “Vihavarta Shofar Teruah” which implies that one must sound a simple blow and then afterwards sound a Teruah as the word “Vihavarta” implies the blowing of a simple sound. After blowing the Teruah one is also required to blow a Tekiah as the verse states “Taviru Shofar” which implies that after blowing the Teruah one is also required to blow a simple blow. This concept is applied by every Teruah whether of Yovel and whether of Rosh Hashanah being that the days are juxtaposed. [Admur 590:1] Thus, in total one is required to blow nine sounds, as the Torah states Teruah three times which teaches one must blow three Teruas and the Torah teaches that every Teruah must have a Tekiah before and after hence equaling nine blows.
 Background of the institution to blow thirty blows: The Torah states that one is to blow a Teruah. The Teruah is translated as a Yebava. The word Yebava is found in the verse regarding the mother of Sisra “The mother of Sisra looked out the window and Tiyabeiv”. This implies that the Yibava is a sound a man makes when he is crying and wailing. There is doubt as to whether this cry is a like the cry of an ill person which gives a slightly lengthy cry [moan-Uhhhhhhh] which consists of several of these sounds [Uhhhh-Uhhhh-Uhhhhh]. This cry is called a Ganach [moan]. Alternatively, perhaps it refers to a person which is crying and wailing very short sounds that come one after the other [Uh-Uh-Uh-Uh-Uh-Uh-Uh]. This sound is called a Yilil [profuse weeping]. Alternatively perhaps it refers to a cry which consist of both sounds one after the other, meaning a Ganach [moan] and then a Yelil, as it is common for one who is weeping to first moan and then cry profusely [Uhhhh-Uhhhh-Uhhhhh-Uh-Uh-Uh-Uh-Uh-Uh-Uh]. [Admur 590:2; See Rosh Hashanah 33b]
Other opinions: Some opinions hold that Biblically all the sounds of the Teruah, whether it be a Teruah or a Shevarim, are considered a valid Teruah, and thus whatever a person sounds, whether only a Shevarim or only a Teruah, he fulfills his [Biblical] obligation. Thus, in earlier generations some communities sounded only the Shevarim and some sounded only the Teruah, and they both fulfilled their Biblical obligation. However since this matter seemed like a dispute in the eyes of the masses therefore the Sages instituted that everyone should blow the same sounds and they should not blow sounds that seem to be under dispute in the eyes of the masses. Hence, they instituted to blow Tashrat; Tashat and Tarat three times each in order to remove the masses from questioning the sounds based on the differences they witness in the customs. [Admur 592:4; based on Rav Haiy Gaon brought in Tosafus 34 that all the sounds are valid]
 As perhaps the Teruah consists of a Genicha and Yelil. [Admur ibid]
 As perhaps the Teruah consists of only a Genicha. Now although one already heard a Genicha in the first set this Genicha was invalidated by the Yelila that followed it as this Yelila intervened between the Genicha and Tekiah that is meant to follow it and the Torah stated that one is to blow a Tekiah after the Teruah without having any sound of a Shofar which is not a Tekiah in between. [Admur ibid]
 As perhaps the Teruah consists of only a Yelil. Now although one already heard a Yelil in the first set this Yelil was invalidated by the Genicha that preceded it as this Genicha intervened between the first Tekiah and Yelil that is meant to follow it. [Admur ibid]
 Admur 592:7
 The original custom was to blow the Shofar before Shacharis as Zerizin Makdimin Lemitzvos. However, this was nullified due to a decree of the gentile monarchy. [Gemara R”H 32b] Hence they would blow the Shofar during Musaf as by that time the government agents were no longer present to witness the blowing. [Rashi ibid] Others however explain that the words of the Gemara “due to the gentile monarchy” means to say that it was dangerous to blow the Shofar before Shacharis as the gentiles thought that we blew the Shofar as a sign of waging war and they would hence attack. The sounds were thus changed to be blown during Musaf as the gentiles will simply think it is part of the regular prayers and not a vouch for war. [Tosafus 32b; Levush 585; Kaf Hachaim 585:5]
 The reason: This is done in order to confuse the heavenly prosecutor prior to the Musaf prayer in order so he does not prosecute during the Musaf prayer, as blowing the Shofar enters the Satan into a state of confusion and distress. [Admur 592:7; Gemara R”H 16b; Rashi there “When the Satan sees how beloved the Mitzvos are to the Jewish people his claims against them are silenced”]
 Siddur Admur; Rambam Shofar 3:10; Maggid Mishneh ibid that so is the custom in all world Jewry; Mentioned in Admur 584:9; Rama 590:9; Birkeiy Yosef 588:1; Machazik Bracha 588:1; Chaim Sheol 2 10:3 in name of Rashash; Shaareiy Teshuvah 588; Mateh Yehuda; Kisei Eliyahu 588:1; Ruach Chaim 588:1; Kaf Hachaim 588:5; M”E 585:1; See Likkutei Sichos 39 p. 43 footnote 1 that the novelty of Admur stating this in the Siddur is to hint to the reason behind blowing the Shofar at this time, which is to remember the sounds that were blown by Matan Torah.
The reason: In order to remember the sound of the Shofar of Matan Torah. [Likkutei Sichos ibid based on Rasag brought in Abudarham] Alternatively based on Kabala the Tikkun of the Shofar is only to be done after Shacharis and the reading of the Torah. [Birkeiy Yosef ibid in name of Mekubalim]
Other Customs: Some have the custom to blow the Shofar before Shacharis as was the original custom [see next]. [Birkeiy Yosef 588:1; Mateh Yehuda; Kisei Eliyahu 588:1 write that this was the custom of their time although he negates the custom and says he would not hear the Shofar before Shacharis.]
 Piskeiy Teshuvos 592:3
 585:5-6; As is the simple implication of the ruling of the Gemara Rosh Hashanah 34b, and Michaber:Admur in 590:1-2 that the 30 blows are Biblically required due to doubt as what consists of a Teruah, and so rules Ramban in Milchamos Hashem. However there are opinions in the Rishonim [Rosh:Chinuch:Ran] which bring Rav Haiy Gaon which rules that in truth only 9 blows are needed and there is no doubt at all, as all the sounds of the Teruah are valid. [This opinion is brought in Admur 592:4]
 Mateh Efraim 586:7 and 601:13 rules that Biblically only 10 blows are required. This ruling is regarding cases [Mudar Hana:Kabalas Shabbos] in which only the mere minimum may be blown and they rule that only 10 sounds are to be blown. So rules also Mishneh Berurah 586:22.
 So explicitly rules the Mateh Efraim and M”B ibid. If however the source of their ruling to blow only 10 times is following Rav Haiy Gaon then seemingly one may blow any combination he wishes, even Tashrat 3x or Tashat 3x, as according to him they are all valid. However Kinyan Torah [3:76] explains that in truth the Mateh Efraim here is suspecting for the opinion of those Rishonim which require all three sounds to be blown, and rather he holds that in truth it is only Rabbinical that three sets of Teruah are required and Biblically one fulfills his obligation simply by blowing one set, hence since there is doubt as to the Teruah we do three sets one of each possibility. This understanding however can certainly not be placed into Admur which clearly rules that the three sets of the Teruah are required Biblically, and hence the blowing of 10 blows would only be allowed if one were to rely on Rav Haiy Gaon, in which case it makes no difference as to which combination one blows three times. [This is unlike the ruling of Piskeiy Teshuvos 590 footnote 2 which says Tashrat may not be sounded 3 times.]
 As there are Poskim [brought next] which rule that one requires 30 blows Biblically.
 Kinyan Torah 3:72
 Kaf Hachaim 586:40; Shraga Hameir 3:11
Regarding the opinion of Admur: It clearly seems that he rules that three sets of Teruas are required Biblically [unlike the suggestion of Kinyan Torah]. However regarding the three types of Teruas that exist there is room to learn that Admur actually rules like Rav Haiy Gaon as a) in 592:4 he mentions this opinion as a matter of fact without dispute [it is only regarding the custom of Musaf that he relates a dispute] and b) after explaining the doubts he writes [unlike Michaber] that the “Sages instituted” to blow 30 times. Now if there is a true Biblical doubt why do we need the Sages to institute this decree, should it not be automatically done due to doubt? Hence this wording seems to imply like Rav Haiy Gaon. Nevertheless since the simple implication of the doubt as written in Admur is that it is a Biblical doubt we therefore wrote above that one is to initially blow 30 blows even if this may compromise on the amount of people he has blown for.
 To properly understand the logistics behind this ruling refer to Halacha 2 [regarding the validity of making a stipulation] and Halacha 9 [regarding blowing other sounds between blows]. Vetzaruch Iyun why this seemingly clear and simple advice is not brought in any previous Poskim! Vetzrauch Iyun Lemaaseh if one should rely on it.
 Heard from Rav Farkash as well as other Rabbanim
 Seemingly it is not necessary for the listener to have this stipulation in mind as it is no different than when the Baal Tokeia intends not to be Motzi someone in which case that person is not Yotzei even if he had in mind. See Chapter 2 Halacha 3.
 Thus if the Shevarim-Teruah is the correct Teruah sound, then he intends to blow them all for the sake of the Mitzvah. If only the Shevarim is the correct Teruah sound then he intends to blow the Shevarim for the Mitzvah and the Teruah is not for the Mitzvah. If only the Teruah is the correct Teruah sound then he intends to blow the Shevarim not for the Mitzvah and the Teruah is being blown for the Mitzvah.
 Kaf Hachaim 585:28 in name of Shaar Hakavanos 99; Peri Eitz Chaim 26:3
 Arizal brought in Minchas Elazar 4:37
The Sages instituted that only one set of תר”ת be sounded after each of the three middle blessings. [592:1] However now [that we have doubt as to the correct sound of the Teruah and therefore] the Sages instituted to blow all the possible sounds of the Teruah in order to avoid the doubt it should have been proper to blow one set of Tashrat Tashat and Tarat by the end of each of the three blessings in order to fulfill one’s obligation according to all the sounds of the Teruah. Nevertheless being that we do not overburden the congregation therefore many are not accustomed to do so. [592:2; See Rambam Shofar 3:12]
 This is based on the background above that one is not required to blow all three sounds of the Teruah three times. Hence various customs evolved regarding which sound of the Teruah is to be preferred when.
 Rama 592:1
 Admur 592:3 [first and Stam opinion]; Rama 592:1; Tur in name of Rabbeinu Tam
The reason: The reason behind this custom is because by blowing the set of Tashrat one sounds every possible sound of the Teruah and hence the only applicable worry is that (perhaps the Teruah is only a Shevarim or only a Teruah and thus) perhaps he is making an interval between the Teruah and the Tekiah that precedes it or proceeds it. Now this alone is of no worry to us being that we already fulfilled our obligation with the Tekios Meyushav, as with those Tekios we fulfilled the obligation according to all the possible doubts. [Admur ibid]
 Admur 592:4; Michaber 592:1 [first opinion]; Rif; Rambam 3:10
 The reason: The reason behind this custom is because [they hold that] Biblically all the sounds of the Teruah, whether it be a Teruah or a Shevarim, are considered a valid Teruah, and thus whatever a person sounds, whether only a Shevarim or only a Teruah, he fulfills his [Biblical] obligation. Thus in earlier generations some communities sounded only the Shevarim and some sounded only the Teruah, and they both fulfilled their Biblical obligation. However since this matter seemed like a dispute in the eyes of the masses therefore the Sages instituted that everyone should blow the same sounds and they should not blow sounds that are under dispute in the eyes of the masses. Hence they instituted to blow Tashrat; Tashat and Tarat three times each in order to remove the masses from questioning the sounds based on the differences they witness in the customs. Accordingly [since blowing all the sounds of the Teruah is only a Rabbinical institution to satisfy the masses] they did not trouble the community to blow all the sounds also by the Tekios of Musaf. On the contrary, with the blows in Musaf we show the public that all the sounds of the Teruah are valid Teruos and a person fulfills his obligation with it. Thus they blow one time Tashrat and one time Tarat and one time Tashat to show that each one of these sets are proper and Kosher. [Admur ibid; based on Rav Haiy Gaon that all the sounds are valid]
 Admur 592:5; Michaber 592:1 [that so is the custom today]; This custom was not mentioned in the Beis Yosef. The Mamar Mordechai 592:1 says that perhaps it came to the Michaber’s attention only after first writing the Beis Yosef.
 Admur 592:5; Aruch; Hamanhig; Ridbaz 29; Riaz; Shalah p. 314; Erech Hashulchan 592:1; Kaf Hachaim 592:4
 Admur ibid; Shlah ibid “praised be the holy congregation that chooses this custom”
The reason: As explained above there is doubt as to which sound is the Teruah, and only in the fourth custom is one for certain blowing a correct set of a Teruah by each blessing. [Shelah ibid]
 Michaber 596:1; first custom brought in Admur ibid; Tur in name of Rav Amram
 The reason: This is done in order to confuse the Satan [Divine prosecutor] from prosecuting that we are not scared of G-d’s judgment. As being that after prayers we proceed to have a festive meal with food drink and merrymaking the prosecution may use this to find fault with us that we are not afraid of the day of judgment. [Thus the final Shofar blowing prevents this prosecution]. [Admur ibid; Taz 596:1] Alternatively the reason is in order to push away the Kelipos from nurturing from the Holiness that was drawn down through the prayer of the day. [Kaf Hachaim 596:1]
 Rama 596:1; Darkei Moshe 596; second custom brought in Admur ibid
 The reason: As they suspect that perhaps some people in the congregation did not properly hear all the blows and hence they will now fulfill their obligation with hearing these blows. [Admur ibid] However the Levush 596 writes that the reason is in order to confuse the Satan. This is similar to the first reason mentioned above. In the Machzor Chabad it states the reason of the Levush that the 30 blows are sounded to confuse the Satan and does not mention the reason of Admur.
 Third custom brought in Admur ibid; M”A 596:1 in name of Shlah based on Arizal in Shaar Hakavanos p. 99; Peri Eitz Chaim 26:3; Hearuch “Meiah Piyos” [that so is their custom]; Hamanhig R”H 21; Tosafus R”H 33b in name of Aruch
 The reason: The reason behind the custom to blow 100 blows is in correspondence to the 100 sounds of cry that the mother of Sisrei sounded. [Hearuch; Hamanhig; Tosafus R”H 33b; brought in Kaf Hachaim 592:1]
 M”B 596:1; M”E 596:1; Kaf Hachaim 596:2; 592:5
How many blows are sounded? There are various customs in regards to how many blows are sounded after the prayer. This depends on how many blows were sounded during prayer. [See Machatzis Hashekel 596:1; Kaf Hachaim 5963] Those communities that don’t blow the Shofar during the silent prayer of Musaf blow 30 blows after Musaf and another 10 blows at the end of Davening, prior to Anim Zemiros. [M”B 592:4; M”E 592:11; Luach Eretz Yisrael] Those communities that blow 30 blows also during the silent Musaf prayer only blow 10 blows after Musaf. These blows are sounded in middle of Kaddish prior to Tiskabel. [Shlah in name of Arizal; Siddur Admur]
A Tekiah Gedola after Tekios Meyushav: It is customary to sound an extra long Tekiah by the concluding blow of the Tekios Meyushav. [Siddur Torah Or and Machzor [omitted in Siddur Slavita of Admur]; Siddur Arizal; Siddur Shlah; Minhagei Mahril R”H; Minchas Elazar 2:65; Darkei Chaim Veshalom 720; Siddur Yaavetz; Siddur Daas Kedoshim; Piskeiy Teshuvos 596:1] The reason this is done is in order so the congregation knows that the blows have ended and they should begin saying Ashrei Haam. [Mahril ibid]
 As we blow 30 by Meyushav; 30 by the silent prayer of Musaf and 30 by the repetition of Musaf for a total of 90. Ten more blows are sounded for a total of 100.
 Siddur Admur; Shlah in name of Arizal
 Siddur Admur; Otzer Minhagei Chabad 334-335
 Sefer Haminhagim p. 120; based on the second custom mentioned above in Admur ibid; This follows the order written in Shlah p. 217 that even after completing the 100 blows as rules the Arizal one is to blow 30 blows after Aleinu. However see Kaf Hachaim 592:5
 In all the new printings of the Machzor Chabad they specifically write that the last Tekiah of the 30 after Tehillim is to be a Tekiah Gedola. [see printing of Machzor Kehos 2006 in both the English and the Hebrew version] However in all previous prints of the Machzor this was omitted and it simply writes “Tashat” as the ending set, without mentioning that it should be a Tekiah Gedola. In Otzer Minhagei Chabad 337 and 335 he evidently understood from this that we do not blow a Tekiah Gedola at the conclusion of the thirty blows after Tehillim.
 Admur 590:18; Bach 596